Data-Generating Process

FCMs as a Measure of Stress

FCM Data

Hunting Events

Movement Data

• Contains the location of the 40 collared deer from Feb. 2020 to Feb. 2023.
• Movement is tracked at hourly intervals.

Limited Data, Large Uncertainty

• Hunting events are single points in time and space.
• Deer locations at hourly intervals \(\Rightarrow\) exact distances unknown \(\Rightarrow\) approximate needed, large uncertainty!
• Each deer only encountered few hunting events.

Limited Data, Large Uncertainty

Limited Data, Large Uncertainty

Other sources of uncertainty include:

• lack of information about hunting events (single time points as start, end, middle?)

• unknown characteristics of the deer (e.g., age, health, etc.),

• other unknown stressors (e.g., predators, human activities, weather, etc.),

• unknown geographical features (e.g., terrain could affect the propagation of sound).

Distance Approximation

Deer location at the time of hunting event is approximated by linear interpolation:

Relevant Hunting Events

The Most Relevant Hunting Event

The Scoring Function

we define the Scoring function as following:

\[ S(d, t) \propto \begin{cases} \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{N}(\mu, \sigma^2) &|t \leq \mu \\ \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{Laplace}(\mu, b) &|t > \mu \end{cases} \] where:

\[ \begin{align*} d & \text{: Distance } \\ t & \text{: Time Difference } \\ \mu & \text{: GRT target = 19 hours } \end{align*} \]

The Scoring Function

The marginal effects of distance and elapsed time since challenge on the score:

The Fused Data

We report models fitted on the following datasets:

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

The Models

We chose two different approaches to Modelling:

1. Statistical Modelling: a model, which helps to understand the effects of our covariates, here a General Additive Mixed Model
2. Machine Learning: a model, which focuses on prediction, in our case a XGBoost Model

A. Generalized Additive Mixed Model

• Family: Gamma

• Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of faecal samples for each deer

  \[ \begin{eqnarray} \textup{FCM}_{ij} &\overset{\mathrm{iid}}{\sim}& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \quad\text{for}\; j = 1,\dots,n_i, \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}), \\ \eta_{ij} &=& \beta_0 + \beta_1 \cdot \textup{number of other relevant hunting events}_{ij} + \\ && f_1(\textup{time difference}_{ij}) + f_2(\textup{distance}_{ij}) + \\ && f_3(\textup{sample delay}_{ij}) + f_4(\textup{defecation day}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\overset{\mathrm{iid}}{\sim}& \mathcal{N}(0, \sigma_\gamma^2) \end{eqnarray} \]

  \(f_1, f_2, f_3, f_4\) are penalized cubic regression splines.

A. Generalized Additive Mixed Model

Main Results

• High uncertainty (large standard error) about estimated effects, in particular of time difference and distance, across all datasets.

• Consistent pattern of sample delay effect when using REML method: larger sample delay \(\Rightarrow\) lower FCM level, as expected.

• Instability with respect to estimation methods. GCV tends to yield more wiggly smooth effects than REML.

• Estimation of random intercepts is sensitive to choice of dataset.

B. XGBoost

XGBoost is a gradient boosting algorithm that builds decision trees sequentially, each one correcting the errors of the previous. It improves accuracy with techniques like regularization, shrinkage, and column subsampling, making it efficient and better at generalization.

It works very well for numerical data and is well implemented which is why we chose it.

XGBoost Workflow

1. Split Data: We divide each dataset into training and testing sets. (75% - 25%)
2. Set Hyperparameter Grid: We define a range of values for starting hyperparameters.
3. Optimize Hyperparameters: We then perform a grid search to evaluate different hyperparameter combinations -> Iteratively readjust the grid based on test RMSE until convergence. Uses CV
4. Train Final Model on Full Data: We then use the optimized parameters to train on the entire dataset. We prevent overfitting by optimal n_rounds to keep Test-RMSE low.
5. Aggregate Results: Run the pipeline 40 times with different seeds. Use the average RMSE and average predictions to evaluate the overall performance.

We do this seperately for all 3 datasets (nearest, closest and score).